C2 chapter 8 - graphs of trigonometric functions

In this chapter you will learn the characteristic shapes of the sine, cosine and tangent graphs relating key points on them to facts derived from an understanding of special triangles. You will learn to transform these graphs by stretching and translating them.

C4 section 8.1

Before you start you need to be able to recall and use the basic definitions of sine, cosine and tangent and be able to work in both degrees and radians.
After completing this section you will have learnt how to use facts about sin, cos and tan of acute angles to find sin, cos and tan of other angles.
Examples:

C2 Exercise 8A

• C2 Exercise 8A worked solutions:
• 

  C2 Exercise 8A (trigonometric functions).jnt
  

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  C2 Exercise 8A (trigonometric functions).pdf
  

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C2 Exercise 8B

• C2 Exercise 8B worked solutions:
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  C2 Exercise 8B (new trig values from old).jnt
  

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  C2 Exercise 8B (new trig values from old).pdf
  

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C4 section 8.2

Before you start you need to be able to recall and use the basic definitions of sine, cosine and tangent and be able to work in both degrees and radians.
After completing this section you will have learnt how to use facts about sin, cos and tan of acute angles to find sin, cos and tan of other angles.
Examples:

C2 Exercise 8C

• C2 Exercise 8C worked solutions:
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  C2 Exercise 8C (CAST diagram and trig relations).jnt
  

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  C2 Exercise 8C (CAST diagram and trig relations).pdf
  

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C4 section 8.3

Before you start you need to be able to recall and use the basic definitions of sine, cosine and tangent ; recall the properties of equilateral and isosceles triangles and recall and be able to apply Pythagoras' theorem.
After completing this section you will be able to calculate exact values for the sine, cosine and tangent of 30°, 45° and 60°. With repeated practice these exact values should be memorized and you should be able to reproduce these proofs on demand.
The corollary to this section is you will be able to calculate exact values for related angles.
Examples:

C2 Exercise 8D

• C2 Exercise 8D worked solutions:
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  C2 Exercise 8D (exact values of trig functions).jnt
  

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  C2 Exercise 8D (exact values of trig functions).pdf
  

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C4 section 8.4

Before you start you need to be able to evaluate the trig functions and plot accurate graphs.
After completing this section you should be able to recognize, recall and visualize the sine, cosine and tangent graphs and describe their symmetry properties.
Examples:

C2 Exercise 8E

• C2 Exercise 8E interactive solutions:
• Sketch the graph of y = cos θ in the interval -π ≤ θ ≤ π
• answer:
• Sketch the graph of y = tan θ in the interval -180 ≤ θ ≤ 180
• answer:
• Sketch the graph of y = sin θ in the interval -90 ≤ θ ≤ 270
• answer:

• Exploring problems similar to this exercise:
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  (insert some Wolfram Widgets or Geogebra pages that plot sin, cos and tan in a variable interval)

C4 section 8.5

Before you start you need to be able to transform graphs. You should review the content of C1 chapter 4.
After completing this section you should be able to perform simple transformations of trig graphs.
Interactive examples:


There are four basic transformations to understand:

	transformation	f(θ) = sin θ	f(θ) = cos θ	f(θ) = tan θ
1.	the horizontal translation through - α $y = \text { f} (θ + α)$
2.	the vertical translation through + a $y = \text { f} (θ) + a$
3.	the horizontal stretch with scale factor 1/n $y = \text { f} (nθ)$
4.	the vertical stretch with scale factor a $y = a \text {f} (θ)$

A particularly tricky topic seem

C2 Exercise 8F

• C2 Exercise 8E interactive solutions: